Complemented subspace

In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space is a vector subspace for which there exists some other vector subspace of called its (topological) complement in , such that is the direct sum in the category of topological vector spaces. Formally, topological direct sums strengthen the algebraic direct sum by requiring certain maps be continuous; the result retains many nice properties from the operation of direct sum in finite-dimensional vector spaces. Every finite-dimensional subspace of a Banach space is complemented, but other subspaces may not. In general, classifying all complemented subspaces is a difficult problem, which has been solved only for some well-known Banach spaces. The concept of a complemented subspace is analogous to, but distinct from, that of a set complement. The set-theoretic complement of a vector subspace is never a complementary subspace. If is a vector space and and are vector subspaces of then there is a well-defined addition map The map is a morphism in the — that is to say, linear. Direct sum and Direct sum of modules The vector space is said to be the algebraic direct sum (or direct sum in the category of vector spaces) when any of the following equivalent conditions are satisfied: The addition map is a vector space isomorphism. The addition map is bijective. and ; in this case is called an algebraic complement or supplement to in and the two subspaces are said to be complementary or supplementary. When these conditions hold, the inverse is well-defined and can be written in terms of coordinates as The first coordinate is called the canonical projection of onto ; likewise the second coordinate is the canonical projection onto Equivalently, and are the unique vectors in and respectively, that satisfy As maps, where denotes the identity map on . CoproductDirect sum#Direct sum in categories and Direct sum of topological groups Suppose that the vector space is the algebraic direct sum of .